A321456 Numbers k that are divisible by sum(pi)^2+sum(ei) where k=p1^e1*...*pj^ej with pi primes.
16, 192, 288, 704, 1470, 2112, 2160, 3168, 3240, 3872, 4096, 4608, 4752, 4860, 5400, 6912, 7128, 7245, 8100, 9295, 10368, 11616, 13500, 15552, 15900, 17424, 21296, 23328, 23850, 26136, 27720, 32830, 34992, 35960, 39600, 39750, 41536, 45584, 52250, 52488, 59400, 62920, 63888, 67200, 78732, 81920, 86430
Offset: 1
Keywords
Examples
704 is an item as its prime factorization is 2^6+11^1, sum(pi)=2+11=13, sum(e1)=6+1=7, sum(pi)^2+sum(e1)=13^2+7=169+7=176, finally 704=c*176 for c=4.
Programs
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Mathematica
fun[n_] := Module[{f = FactorInteger[n]}, Total@f[[;;, 1]]^2 + Total@f[[;;, 2]]]; aQ[n_] := Divisible[n, fun[n]]; Select[Range[100000], aQ] (* Amiram Eldar, Nov 18 2018 *) -
PARI
ok(k)={my(f=factor(k)); k > 1 && k % (vecsum(f[,2]) + vecsum(f[,1])^2) == 0} \\ Andrew Howroyd, Nov 19 2018 -
Python
from sympy.ntheory import factorint, isprime n=100000 r="" def calc(n): global r a=factorint(n) lp=[] for p in a.keys(): lp.append(p) lexp=[] for exp in a.values(): lexp.append(exp) if n%((sum(lp))**2+sum(lexp))==0: r += "," r += str(n) return for i in range(4,n): calc(i) print(r[1:])
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